Understanding Engineering Mathematics

4.3.7 Some applications of logarithms

➤➤

120 134

➤

A.Solve the following equations, giving your answers to 3 decimal places.

(i) 3x= 16 (ii) 4^2 x= 9 (iii) 4× 5 −^2 x= 3 × 7 x−^2 (iv) 3x= 42 x−^1

B.Convert the following equations to straight line form

(i) y= 4 x^7 (ii) y= 3 x−^4 (iii) y=

5

x^3

(iv) y= 20 e−^2 x (v) y= 24 x−^1

4.4 Applications

You will see many applications of the exponential function in later chapters, particularly

in complex numbers, differential equations and the Laplace transform.

1.Thehyperbolic functionscosh, sinh, tanh are defined by

coshx=

ex+e−x

2

(hyperbolic cosine)

sinhx=

ex−e−x

2

(hyperbolic sine)

tanhx=

sinhx

coshx

(hyperbolic tan)

These are frequently occurring functions in engineering – for example the shape of a

cable suspended at two ends can be described by thecatenary, which is essentially the

coshxcurve. The hyperbolic functions obey very similar identities to the trig functions.

In particular, show that

(i) cosh^2 x−sinh^2 x= 1 (cosh^2 x=(coshx)^2 ,etc.).

(ii) sinh(A+B)=sinhAcoshB+sinhBcoshA.

Use trig identities to suggest other hyperbolic identities and use the above definitions

to confirm them.

(iii) Evaluate cosh 0, sinh 0.

(iv) Given that

d

dx

(ex)=exand

d(ex)

dx

=−e−xdeduce the derivatives of sinhxand

coshx.

(v) Plot the graphs of sinhxand coshx.

2.The current/voltage characteristic of a rectifying contact in a semiconductor device is

given by

I=Io

[

exp

(

eV

kT

)

− 1

]

At room temperaturekT/eis about 25 mV. Plot the curve ofI/IoforVbetween− 60

and 60 mV in this case. Show that for values ofVgreater than 75 mV the increase in